A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B

Q: Why do the individual pieces move in different directions if the CM doesn't shift?

Internal forces push the pieces apart, giving them equal and opposite momenta relative to the CM. However, because these forces cancel out for the system as a whole, the CM itself remains completely unaffected.

Q: What if one piece is much heavier than the other?

The heavier piece will naturally stay closer to the CM, while the lighter piece will be pushed further away. This perfectly balances their mass distribution, ensuring the CM stays exactly on its original path.

Q: Does air resistance change this outcome?

Yes, if we consider air resistance, it acts as an external force. The change in shape after breaking might alter the total air resistance, which would then affect the CM. But in standard physics problems, we ignore air resistance unless explicitly stated.

Q: What if the body was thrown horizontally instead of dropped?

The CM would follow a parabolic projectile path. Even after breaking mid-air, the CM would continue to trace that exact same parabola as if the explosion never happened.

Q: Is kinetic energy conserved during the breaking process?

No, the internal explosion releases energy (like chemical energy turning into kinetic energy), so the total kinetic energy of the pieces is greater than the original body. However, the momentum of the CM is strictly conserved.

Question

A body $A$ of mass $M$ while falling vertically downwards under gravity breaks into two parts; a body $B$ of mass $\frac{1}{3}M$ and a body $C$ of mass $\frac{2}{3}M$. The centre of mass of bodies $B$ and $C$ taken together shifts compared to that of body $A$ towards

Options:

(A) depends on height of breaking

(B) does not shift

(C) body $C$

(D) body $B$

Sigma Percentile · Accepted Answer

### The Illusion of the Explosion

Imagine a firecracker falling from the sky. Suddenly, it explodes into multiple fragments flying in all directions.

It seems chaotic, doesn't it? You might think the overall path of the firecracker is completely ruined.

But physics tells us a different, beautifully elegant story.

### The Master of the System: Centre of Mass

To understand this, we must look at the system as a whole. The motion of the entire system is represented by a single special point: the **Centre of Mass (CM)**.

According to Newton's Second Law for a system of particles, the acceleration of the CM is dictated strictly by the net **external force**.

$$ \vec{F}_{ext} = M \vec{a}_{CM} $$

Internal forces, no matter how violent or explosive, always occur in equal and opposite pairs. They cancel each other out perfectly and have absolutely zero effect on the CM.

### Analyzing the Falling Body

In our problem, body $A$ is falling freely under gravity. The only external force acting on it is its weight, $Mg$.

Therefore, its initial acceleration is simply $g$ downwards.

When it breaks into bodies $B$ and $C$, the forces causing this split are purely internal.

### The Unchanged Path

After the split, what are the external forces acting on the fragments? Gravity is still pulling down on body $B$ with force $m_B g$ and on body $C$ with force $m_C g$.

The total external force remains $(m_B + m_C)g = Mg$.

Since the net external force hasn't changed, the acceleration of the CM remains exactly $g$ downwards.

The fragments may fly apart, but their **Centre of Mass** continues to fall along the exact same straight vertical line as if the body never broke at all!

This is a classic and favorite concept in JEE. Always remember: internal forces can rearrange the pieces, but they can never alter the destiny of the Centre of Mass.

Select Answer:

Visualized Solution

$a_{C M} before breaking$

$The Governing Law$

$Analyzing the Explosion$

$Net External Force$

$Final Acceleration$

$The Way Forward$

The Sigma Insight: Motion of Centre of Mass

The Illusion of the Explosion

The Master of the System

Analyzing the Falling Body

The Unchanged Path

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } aCM​ before breaking

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Governing Law

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Explosion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Net External Force

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Acceleration

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward